Abstract
The mechanisms by which nonrandom mating affects selected populations are not completely understood and remain a subject of scientific debate in the development of tractable predictors of population characteristics. The main objective of this study was to provide a predictive model for the genetic variance and covariance among mates for traits subjected to directional selection in populations with nonrandom mating based on the pedigree. Stochastic simulations were used to check the validity of this model. Our predictions indicate that the positive covariance among mates that is expected to result with preferential mating of relatives can be severely overpredicted from neutral expectations. The covariance expected from neutral theory is offset by an opposing covariance between the genetic mean of an individual's family and the Mendelian sampling term of its mate. This mechanism was able to predict the reduction in covariance among mates that we observed in the simulated populations and, in consequence, the equilibrium genetic variance and expected longterm genetic contributions. Additionally, this study provided confirmatory evidence on the postulated relationships of longterm genetic contributions with both the rate of genetic gain and the rate of inbreeding (ΔF) with nonrandom mating. The coefficient of variation of the expected gene flow among individuals and ΔF was sensitive to nonrandom mating when heritability was low, but less so as heritability increased, and the theory developed in the study was sufficient to explain this phenomenon.
RECENT advances in quantitative genetic theory have allowed breeding schemes to consider the management of genetic variation objectively, simultaneously with the maximization of genetic gain. Such advances are highly relevant to breeding practice, not only for commercial schemes but also for those schemes that are orientated toward the conservation of genetic resources. These advances include the development of tractable, deterministic predictors of rates of inbreeding (ΔF), gene flow, and genetic gain (ΔG) for complex selected populations (Woolliamset al. 1999; Woolliams and Bijma 2000), and operational tools for daytoday selection decisions (Wray and Goddard 1994; Meuwissen 1997; Grundyet al. 1998).
However, the theory underpinning these tools has been developed primarily for random mating of the selected males and females and for a neutral locus where the genotypic frequencies in the offspring display no departure from HardyWeinberg (HW) proportions, other than that arising from the partitioning of the gene pool induced by two sexes (Robertson 1965). However, the role of nonrandom mating, where some specific design is placed upon which male is mated to which female, has been less well studied in this context. Although utilizing a mating design to minimize ΔF does not always lead to substantial deviations from HW proportions (Sonesson and Meuwissen 2000), much of the theory on genetic variation and the impact of nonrandom mating has been built up around the concept of departures from HW equilibrium (e.g., Caballero and Hill 1992; Santiago and Caballero 1995; Wang 1996) and the concepts of the avoidance of, or preference for, mating relatives.
The interpretation of the work on nonrandom mating, both its application and its impact, remains a subject of scientific debate. For example, in conservation, accepted practice uses minimum coancestry to minimize ΔF (Caballeroet al. 1996; Frankhamet al. 2002), yet recent theoretical developments using genetic contributions show that the lowest ΔF with hierarchical matings is achievable when relatives are preferentially mated (Sánchezet al. 2003). In selected populations, there is clear evidence that mating designs are beneficial, although not all these designs define matings through pedigree, e.g., factorial mating (Woolliams 1989). Several articles indicate that attention to the pedigree relationships within a mating design can be advantageous over randommating schemes (e.g., Santiago and Caballero 1995; Caballeroet al. 1996; Sánchezet al. 1999). However, the benefits of these methods have not always been quantified in predictable terms for their joint impact upon genetic gain and rates of inbreeding. Moreover, other studies on mating designs (Toro and PérezEnciso 1990; Klieveet al. 1994; Fernández and Toro 1999) do not separate out the impact of specifying the matings from the impact of controlling only the overall contribution of each selected individual assuming random mating.
Santiago and Caballero (1995) were successful in producing predictions of the effective population size [N_{e}, defined as (2ΔF)^{–1}] in nonrandom mating populations undergoing mass selection, using an approach based upon the variance of allele frequencies. These authors considered nonrandom mating in the form of departures from HW equilibrium achieved through partial fullsib mating. However, Woolliams and Bijma (2000) showed that, for random mating, an approach to predicting N_{e} using longterm genetic contributions was capable of extension to more complex selection schemes. To achieve such an extension with nonrandom mating requires an understanding of the expected gene flows of individuals based upon the inheritance of selective advantages, which in turn presupposes an understanding of the behavior of the genetic variation. There appears to be little published information on these topics.
Therefore, the main objective of this study is to advance the theoretical framework for predicting the impact of nonrandom mating for populations undergoing directional selection, providing a predictive model for the genetic variance and covariance among mates. The nonrandom mating is defined solely in terms of departures from HW proportions for neutral alleles, without reference to phenotypes. The impact is measured in terms of genetic parameters, such as genetic variance, expected gene flow, and ΔF, assuming the infinitesimal model and mass selection. In the course of this article, the opportunity to validate the developments of Woolliams and Bijma (2000) for nonrandom mating is taken. The accuracies of the predictions of ΔF and ΔG and the validity of the framework are established with the help of stochastic simulations.
MATERIALS AND METHODS
Nonrandom mating and neutral theory: The correlation between uniting gametes due to the nonrandom mating of parents is an additional factor affecting heterozygosity over and above initial gene frequencies and their accumulated drift. Using the classical Fstatistics of Wright (1969), the expected fractional decrease in the heterozygosity for a given population (F_{IT}) can be related to two further statistics using the relationship (1 – F_{IT}) = (1 – F_{IS})(1 – F_{ST}), where F_{ST} is the fractional loss of heterozygosity due to the finiteness of the population census, and F_{IS} is the loss of heterozygosity due to the nonrandom mating of parents. The F_{IS} can be seen as a correlation of gene effects between homologous alleles in pairs of mating parents, i.e., the correlation between alleles within infinitely many conceptual offspring derived from each of the pairs of mating parents (often denoted α_{O} and hereafter in this study). Whereas, if conceived of as the correlation between alleles within successful offspring (i.e., existing individuals), it measures the actual loss of heterozygosity due to the nonrandomness of the mating of the parents with nonzero contributions in the offspring generation (often denoted as α_{I}). Thus the former (α_{O}) is a potential correlation of uniting gametes from selected parents, whereas the latter (α_{I}) is a realized correlation and will be affected by the finite random sampling of gametes to form the offspring generation and by any artificial and/or natural selection of offspring before reaching the breeding population.
Under the assumption of completely neutral genes α_{O} would tend to be equal to α_{I}. For neutral genes, when nonrandom mating results from a mix of fullsib mating and random mating, α_{I} (i.e., α_{I} ≈ α_{O} =α) can be related to the proportion of fullsib mating (denoted hereafter φ)by α= φ/(4 – 3φ)(Ghai 1969).
To obtain α_{O} in a breeding population, let θ_{ij} be the coancestry (kinship) coefficient (Lynch and Walsh 1998) of sire i and dam j, then following Wright's (1969) equation
To obtain a summary value of α_{I} for a pedigreed population undergoing selection, we follow the same reasoning as above. Note that
The terms α_{I(}_{k}_{)} are used to define two related, but distinct, summary α_{I}: (i) α_{I}_{c} = Σ_{k}c_{k}α_{I(}_{k}_{)}, where c_{k} was the observed contribution of k to the selected offspring in the next generation, and (ii) α_{I}_{r} = Σ_{k}r_{k} α_{I(}_{k}_{)}, where r_{k} was the longterm genetic contribution (described in the following sections) of k. If we assume now that directional selection has taken place among families, then the direct equivalence between α_{O} and α_{I} (irrespective of whether Ic or Ir) no longer holds for selective genes or neutral genes since selection success will depend on the α_{I} of the parents (Caballeroet al. 1996).
Dynamics of genetic (co)variance for a selected trait under nonrandom mating: In this section, a model is developed to show that the impact of nonrandom mating on covariances between mates for selected traits and neutral traits may be qualitatively different and to describe the circumstances under which this can occur. In particular, it demonstrates that selection induces a negative covariance between true family means and Mendelian sampling terms, not only within individual selected parents but also between a parent and its mate, thereby reducing genetic variance more than would be predicted by previously existing selection theory.
Consider a population with equal numbers of dams and sires, i.e., a mating ratio of 1, mated in pairs to produce a deviation from HW equilibrium equal to α_{O} on the basis of pedigree information alone. For the trait under selection, assume a heritability of
Let P_{i} denote the phenotype of an individual i for the selected trait and B_{i} be the breeding value for a neutral trait, then the covariance between breeding values of mates is given by Caballero and Hill (1992) as
When allocating mates (i, j) using the pedigree alone,
Since
Consequently, for the selected trait, the total observed additive variance may decline even as α_{O} increases (demonstrated in the results), although α_{O} is superficially increasing one component of the variation. This is a phenomenon associated with linkage disequilibrium and arises from (i) a lower Mendelian sampling variance replenishing the genetic variation lost due to selection in each generation and (ii) the induction of negative covariance between the Mendelian sampling term of a parent and the true family mean of its mate.
Predictions of rate of inbreeding and genetic gain through the concept of longterm genetic contributions: The genetic contribution of an ancestor k (denoted r_{k}) to a descendant j is the proportion of genes carried by j that are expected to derive by descent from the ancestor k. A descendant's breeding value can be decomposed into a sum of Mendelian sampling deviations from all ancestors, with the weighting for ancestor k's Mendelian deviation being r_{k}. For a mixing population, after a sufficiently large number of generations, k's genetic contribution to all individuals within the population approaches the same stable and constant value across generations. In the remainder of the text, the stable genetic contributions from distant ancestors are referred to as “longterm genetic contributions.” The longterm genetic contributions will reflect differences among individual ancestors arising from their respective selective advantages together with cumulative chance factors across generations. Therefore, longterm genetic contributions model the gene flow of individual ancestors through the population.
The asymptotic ΔF for nonrandom mating can be derived through its theoretical relationship with the sum of squared longterm genetic contributions,
For mass selection, the selective advantages for an ancestor are its own breeding value and those of its mates. This set of selective advantages influences not only the breeding success of the resulting offspring from that given ancestor, but also that of subsequent descendants. This dependence of the gene flow on the selective advantage can be expressed as a conditional expectation (μ_{k}), i.e., as a function of the selective advantages. For truncation selection based upon phenotype, μ_{k} can be satisfactorily modeled as a linear relationship between the genetic contribution and the breeding values,
Population model and procedures for stochastic simulation: This section describes the general population model and selection procedures for which predictions and simulations will be compared. The population was reproduced in discrete generations with a constant breeding size of N_{S} sires and N_{D} dams and a mating ratio of dams to sires of 1 in all generations (N_{S} = N_{D} = N). Each dam mothered n_{O} offspring, all fullsibs, and comprising equal numbers of male and female candidates. Selection was upon phenotype P, which was the sum of breeding value A and an environmental deviation.
For simulation, a noninbred and unrelated base population was generated with
Selected individuals were mated following a mating design with nonrandom mating based upon α_{O} (as defined earlier), which was carried out systematically in all generations except in the base population, where founders were randomly allocated in pairs. The allocation of mates was decided in such a way that α_{O} was as close as possible to a target value α_{Ofix}. This process involved a search throughout the feasible set of matings, carried out by the simulated annealing technique (Presset al. 1992), to minimize the objective function given by (α_{O} –α_{Ofix})^{2}. A random sample of matings was used as a starting point. The maximum feasible value for α_{O} is 1, which can be attained by multiple generations of close inbreeding leading to sublining. For the benefit of a more general scheme, the upper limit of α_{Ofix} was <1 in this study. On the other extreme, the lowest possible value of α_{O} in finite populations lies much closer to what is expected for randommating populations, due to the fact that the avoidance of inbreeding is constrained in the long term by the genetic depletion caused by drift, as pointed out by Caballero and Hill (1992). The values of α_{Ofix} used in the simulation were –0.03, 0, 0.03, 0.06, 0.12, 0.18, and 0.24.
Longterm genetic contributions were calculated for an ancestral generation born after 20 generations of selection from the unselected base and upon the cohort of descendants born 20 generations after that ancestral generation. This guaranteed attainment of equilibrium of genetic variances in all the cases with α_{Ofix} ≤ 0.12. With more extreme α_{Ofix}, however, a longer period of time was needed before such equilibrium is reached (Santiago and Caballero 1995). For these cases, 20 further generations were bred before establishing the ancestral generation, although it was found unnecessary to extend the period of time for obtaining summary statistics for the converged contributions (i.e., 20 generations from ancestors to descendants). Observed longterm genetic contributions were used to calculate the predictions of ΔF and ΔG from Equations 6 and 7.
The values of achieved α_{O}, α_{I}_{c}, α_{Ir}, and
RESULTS
Expected vs. observed degree of nonrandom mating: The degree of nonrandom mating is described in this section, in terms of (i) the two distinct summary α_{I}'s (i.e., α_{I}_{c} and α_{I}_{r}) vs. α_{Ofix} and (ii) the expected and observed proportion of fullsib matings (φ_{exp} and φ_{obs}, respectively).
For item i, Table 1 shows that substantial deviations between α_{I}_{r} and α_{Ic} occurred, with α_{I}_{r} <α_{I}_{c} as
For item ii, Figure 1 shows that high values of α_{Ofix} led to important deviations between observed and expected values of φ, with φ_{obs} < φ_{exp}, although a good fit was obtained for intermediate α_{Ofix}. Note that the simulations were implemented through general algorithms for nonrandom mating so that α_{O} was attained through multiple sources of nonrandomness rather than through fullsib mating alone, although since N_{S} = N_{D} there were no halfsibs. For the lower extreme shown in Figure 1, with α_{Ofix} = 0, φ_{obs} was >0 although φ_{exp} = 0. This should be expected for two reasons: in a small population the probability of a fullsib mating is not vanishingly small as is explicitly assumed in the result of Ghai (1969), and random allocation of mates with two sexes will result in a marginally negative α_{O} (Robertson 1965), requiring some fullsib matings in compensation.
Effects of nonrandom mating on the genetic (co)variances for the selected trait: In this section, we describe the effects of nonrandom mating on the selected trait for (i) the genetic covariance among mates and (ii) the genetic variance. Simulated genetic covariance among mates is shown in Figure 2 for a range of α_{Ofix} and
Given that the genetic covariance among mates contributes to the genetic variance under nonrandom mating, a reduction in the former component from that predicted by neutral theory will potentially result in a reduction in the latter. This is confirmed in Figure 3 with stochastic simulations and predictions using Equation 5. With selection, the genetic variance in the population can be lower with α_{Ofix} > 0 than when comparable selection is practiced in randomly mated populations. The predictions from Equation 5 tend to overpredict the genetic variance by more than is expected from reductions due to finite sample size alone.
Effects of nonrandom mating on the expected gene flow: Figure 4 shows the relationship between the regression coefficient of longterm genetic contributions on the sum of selective advantages of mating pairs and α_{Ofix}. For a given α_{Ofix}, for all combinations of
It may be expected from neutral theory that increasing α_{O} would increase the regression on the selective advantage since selected offspring will be more likely to be mated to relatives, so reinforcing the strength or weakness of the inherited selective advantage, i.e., the mean parental breeding value, not only predicts the breeding value of its offspring but also predicts that of its offspring's mate. This is clearly the case for low
The impact of the nonrandom mating on the expected gene flows conditional on the selective advantage (i.e., the sum of the breeding values of an individual and its mate) is shown in Figure 5, measured by the CV of μ_{k} (see Equation 10). The values presented in Figure 5 use parameters in Equation 10 estimated from the simulations. For α_{Ofix} = 0, it is clear that the impact of the selective advantage on gene flow is greatest when
Effect of nonrandom mating on predictions of ΔF and ΔG based on longterm genetic contributions: The effect of nonrandom mating on ΔF is shown in Figure 6 and Table 2 contrasts predictions using Equation 6 for two different selection intensities. The pattern of relationship between ΔF and
Predictions of ΔF using Equation 6 always underestimated the observed ΔF, but this is expected by a fraction approximately equal to 2ΔF (Woolliams and Bijma 2000). When this is accounted for (as in Table 2), the serious errors occur only when selection intensity and α_{Ofix} are high. The pattern of these errors is similar to the cases in Table 1, where α_{I}_{c} and α_{I}_{r} show serious discrepancies. The predictions shown use α_{I}_{r} in Equation 6, and not α_{I}_{c}, since α_{I}_{r} provided more reliable predictions than α_{I}_{c}. Where serious discrepancies occurred between the observed ΔF and ΔF predicted from Equation 6, the prediction error could be approximately halved (results not shown) by modifying Equation 6 to be
DISCUSSION
This article has provided a novel model for predicting the impact of nonrandom mating on the covariance among mates of populations undergoing selection. Examination of the predictions obtained from this model showed that extrapolating expectations of genetic variance and covariance among mates for a neutral trait with nonrandom mating can be qualitatively wrong, with deviations toward severe overprediction. Deviations were largest when heritability and selection intensity were large and there was a strong preferential mating of relatives. While nonrandom mating had a considerable effect upon the impact of selective advantage for low heritability, as measured by the regression of genetic contributions on the selective advantage and the CV of the expected gene flow conditional on the selective advantage, the phenomenon described by the model substantially reduces this effect for moderate heritabilities. Furthermore, the study showed that high selection intensity can induce a negative covariance between the longterm genetic contribution of an ancestor and its α_{I}, particularly when α_{O} is large, and suggested that selection acts to attenuate the strong preferential mating of relatives.
A logical starting point for interpreting the results of deviations from neutral expectations is the genetic covariance achieved among mates for a selected trait when nonrandom mating was practiced. Naively, the preferential mating of relatives would be expected to result in a clear positive genetic covariance among breeding values, since for a neutral trait this covariance has an expectation equal to
The mechanism underlying this model was potentiated as the intensity of selection increased and as the heritability increased. In this article where the results presented have been concerned with selection upon phenotype, the heritability represented the squared accuracy of selection and, together with the value of α_{I}, determined the split in information between the pedigree and the Mendelian sampling term (important in Equations 2, 3, and 5). In more general selection schemes the power of the mechanism would depend on the balance of pedigree information on a candidate and information on its Mendelian components and the use made of such information (e.g., withinfamily selection should not generate such a mechanism), rather than on the accuracy alone.
While the model provides an explanation of some of the results, it has some limitations. First, while its predictions are more credible than those based on neutral theory, the precision leaves some scope for believing that other mechanisms may be operating. Of greater significance is that the covariance between mates is estimated by assuming that the proportion of fullsib mating was that predicted by Ghai's (1969) formula. The use of this formula has two problems: (i) it is limited to schemes with equal numbers of males and females since it cannot cope with nonrandomness coming from other sources such as preference/avoidance of halfsibs and (ii) the predictions provided by Ghai while broadly reliable were not without error. In the model, Ghai's formula was used to translate the desired α_{O} to an expected covariance among the true family means of mates in the selected population; consequently, some improvement might arise from a more general approach to this relationship.
The reduction in covariance between mates arising with selection has direct consequences for the additive genetic variance and for the relationship between longterm genetic contributions and the selective advantage. Both are reduced below expectations based upon neutral theory. The impact on genetic variance is sufficient for the equilibrium genetic variance (i.e., where Mendelian sampling variance is not reduced each generation as inbreeding progresses) to be less for preferential mating of relatives than for random mating when
The impact of
In conclusion, this study has described mechanisms that influence the covariance observed between mates for a trait that is subject to selection when mating is nonrandom. In particular, the covariance is substantially less than that expected from neutral theory, particularly when the heritability is moderate or high, and this has consequences for the observed additive genetic variance, the scale of expected gene flow that is directly attributable to the selective advantage, and ΔF. The observed sensitivity to nonrandom mating of the latter two phenomena when heritability is low can be explained with reference to neutral theory, and the study shows that it is the lack of sensitivity for moderate to high heritabilities that required the development of theory.
Acknowledgments
We are grateful to two anonymous referees whose careful reading and constructive suggestions greatly improved all aspects of this manuscript. The authors gratefully acknowledge support from the European Commission in the form of a Marie Curie Fellowship (L.S.) and from the Department for the Environment, Food, and Rural Affairs in the United Kingdom (J.A.W.).
Footnotes

Communicating editor: S. W. Schaeffer
 Received March 19, 2002.
 Accepted September 19, 2003.
 Copyright © 2004 by the Genetics Society of America